67.7.12 problem Problem 5(d)

Internal problem ID [14052]
Book : APPLIED DIFFERENTIAL EQUATIONS The Primary Course by Vladimir A. Dobrushkin. CRC Press 2015
Section : Chapter 8.3 Systems of Linear Differential Equations (Variation of Parameters). Problems page 514
Problem number : Problem 5(d)
Date solved : Monday, March 31, 2025 at 12:01:28 PM
CAS classification : system_of_ODEs

\begin{align*} \frac {d}{d t}x \left (t \right )&=-7 x \left (t \right )+4 y \left (t \right )+6 \,{\mathrm e}^{3 t}\\ \frac {d}{d t}y \left (t \right )&=-5 x \left (t \right )+2 y \left (t \right )+6 \,{\mathrm e}^{2 t} \end{align*}

With initial conditions

\begin{align*} x \left (0\right ) = 1\\ y \left (0\right ) = -1 \end{align*}

✓ Maple. Time used: 0.174 (sec). Leaf size: 57

ode:=[diff(x(t),t) = -7*x(t)+4*y(t)+6*exp(3*t), diff(y(t),t) = -5*x(t)+2*y(t)+6*exp(2*t)]; 
ic:=x(0) = 1y(0) = -1; 
dsolve([ode,ic]);
 

\begin{align*} x \left (t \right ) &= \frac {44 \,{\mathrm e}^{-3 t}}{5}-\frac {46 \,{\mathrm e}^{-2 t}}{5}+\frac {{\mathrm e}^{3 t}}{5}+\frac {6 \,{\mathrm e}^{2 t}}{5} \\ y \left (t \right ) &= -{\mathrm e}^{3 t}+\frac {44 \,{\mathrm e}^{-3 t}}{5}-\frac {23 \,{\mathrm e}^{-2 t}}{2}+\frac {27 \,{\mathrm e}^{2 t}}{10} \\ \end{align*}

✓ Mathematica. Time used: 0.061 (sec). Leaf size: 66

ode={D[x[t],t]==-7*x[t]+4*y[t]+6*Exp[3*t],D[y[t],t]==-5*x[t]+2*y[t]+6*Exp[2*t]}; 
ic={x[0]==1,y[0]==-1}; 
DSolve[{ode,ic},{x[t],y[t]},t,IncludeSingularSolutions->True]
 

\begin{align*} x(t)\to \frac {1}{5} e^{-3 t} \left (-46 e^t+6 e^{5 t}+e^{6 t}+44\right ) \\ y(t)\to \frac {1}{10} e^{-3 t} \left (-115 e^t+27 e^{5 t}-10 e^{6 t}+88\right ) \\ \end{align*}

✓ Sympy. Time used: 0.235 (sec). Leaf size: 63

from sympy import * 
t = symbols("t") 
x = Function("x") 
y = Function("y") 
ode=[Eq(7*x(t) - 4*y(t) - 6*exp(3*t) + Derivative(x(t), t),0),Eq(5*x(t) - 2*y(t) - 6*exp(2*t) + Derivative(y(t), t),0)] 
ics = {} 
dsolve(ode,func=[x(t),y(t)],ics=ics)
 

\[ \left [ x{\left (t \right )} = \frac {4 C_{1} e^{- 2 t}}{5} + C_{2} e^{- 3 t} + \frac {e^{3 t}}{5} + \frac {6 e^{2 t}}{5}, \ y{\left (t \right )} = C_{1} e^{- 2 t} + C_{2} e^{- 3 t} - e^{3 t} + \frac {27 e^{2 t}}{10}\right ] \]